TMUA 2020 D513/02: Questions & Worked Solutions

Question 1

1 mark

Find the complete set of values of

k

for which the line

y = x - 2

crosses or touches the curve

y = x^2 + kx + 2

A. $-1 \le k \le 3$
B. $-3 \le k \le 5$
C. $-4 \le k \le 4$
D. $k \le -1$ or $k \ge 3$
E. $k \le -3$ or $k \ge 5$
F. $k \le -4$ or $k \ge 4$

Answer: E

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Question 2

1 mark

Given that

\tan \theta = 2

and

180^\circ < \theta < 360^\circ

, find the value of

\cos \theta

A. $\sqrt{3}$
B. $-\sqrt{3}$
C. $\frac{\sqrt{3}}{2}$
D. $-\frac{\sqrt{3}}{2}$
E. $\frac{\sqrt{5}}{5}$
F. $-\frac{\sqrt{5}}{5}$
G. $\frac{2\sqrt{5}}{5}$
H. $-\frac{2\sqrt{5}}{5}$

Answer: F

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Question 3

1 mark

A student makes the following claim:
For all integers

n

, the expression

4\left(\frac{9n+1}{2} - \frac{3n-1}{2}\right)

is divisible by 3.
Here is the student's argument:

4\left(\frac{9n+1}{2} - \frac{3n-1}{2}\right)=2\left(2\left(\frac{9n+1}{2} - \frac{3n-1}{2}\right)\right)

(I)

= 2(9n + 1 - 3n - 1)

(II)

= 2(6n)

(III)

= 12n

(IV)

= 3(4n)

(V)
which is always a multiple of 3. (VI)
So the expression

4\left(\frac{9n+1}{2} - \frac{3n-1}{2}\right)

is always divisible by 3.
Which one of the following is true?

A.The argument is correct.
B.The argument is incorrect, and the first error occurs on line (I).
C.The argument is incorrect, and the first error occurs on line (II).
D.The argument is incorrect, and the first error occurs on line (III).
E.The argument is incorrect, and the first error occurs on line (IV).
F.The argument is incorrect, and the first error occurs on line (V).
G.The argument is incorrect, and the first error occurs on line (VI).

Answer: C

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Question 4

1 mark

Consider the following statement:
Every positive integer

N

that is greater than 6 can be written as the sum of two non-prime integers that are greater than 1.
Which of the following is/are counterexample(s) to this statement?
I

N = 5

II

N = 7

III

N = 9

A.none of them
B.I only
C.II only
D.III only
E.I and II only
F.I and III only
G.II and III only
H.I, II and III

Answer: G

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Question 5

1 mark

Which one of the following shows the graph of

y = \frac{2^x}{1 + 2^x}

(Dotted lines indicate asymptotes.)

A.Graph A
B.Graph B
C.Graph C
D.Graph D
E.Graph E
F.Graph F

Answer: A

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Question 6

1 mark

The function

f(x)

is defined for all real values of

x

.
Which of the following conditions on

f(x)

is/are necessary to ensure that

\int_{-5}^0 f(x)\,dx = \int_0^5 f(x)\,dx

Condition I:

f(x) = f(-x)

for

-5 \le x \le 5

Condition II:

f(x) = c

for

-5 \le x \le 5

, where

c

is a constant
Condition III:

f(x) = -f(-x)

for

-5 \le x \le 5

A.none of them
B.I only
C.II only
D.III only
E.I and II only
F.I and III only
G.II and III only
H.I, II and III

Answer: A

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Question 7

1 mark

Consider the following conditions on a parallelogram

PQRS

, labelled anticlockwise:
I length of

PQ

= length of

QR

II The diagonal

PR

intersects the diagonal

QS

at right angles
III

\angle PQR= \angle QRS

Which of these conditions is/are individually sufficient for the parallelogram

PQRS

to be a square?

A.Condition I is sufficient: yes, Condition II is sufficient: yes, Condition III is sufficient: yes
B.Condition I is sufficient: yes, Condition II is sufficient: yes, Condition III is sufficient: no
C.Condition I is sufficient: yes, Condition II is sufficient: no, Condition III is sufficient: yes
D.Condition I is sufficient: yes, Condition II is sufficient: no, Condition III is sufficient: no
E.Condition I is sufficient: no, Condition II is sufficient: yes, Condition III is sufficient: yes
F.Condition I is sufficient: no, Condition II is sufficient: yes, Condition III is sufficient: no
G.Condition I is sufficient: no, Condition II is sufficient: no, Condition III is sufficient: yes
H.Condition I is sufficient: no, Condition II is sufficient: no, Condition III is sufficient: no

Answer: H

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Question 8

1 mark

A student is asked to prove whether the following statement (*) is true or false:
(*) For all real numbers

a

and

b

,

|a + b| < |a| + |b|

The student's proof is as follows:
Statement (*) is false. A counterexample is

a = 3

,

b = 4

, as

|3 + 4| = 7

and

|3| + |4| = 7

, but

7 < 7

is false.
Which of the following best describes the student's proof?

A.The statement (*) is true, and the student's proof is not correct.
B.The statement (*) is false, but the student's proof is not correct: the counterexample is not valid.
C.The statement (*) is false, but the student's proof is not correct: the student needs to give all the values of $a$ and $b$ where $|a + b| < |a| + |b|$ is false.
D.The statement (*) is false, but the student's proof is not correct: the student should have instead stated that for all real numbers $a$ and $b$ , $|a + b| \le |a| + |b|$ .
E.The statement (*) is false, and the student's proof is fully correct.

Answer: E

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Question 9

1 mark

A student wishes to evaluate the function

f(x) = x \sin x

, where

x

is in radians, but has a calculator that only works in degrees.
What could the student type into their calculator to correctly evaluate

f(4)

?

A. $(\pi \times 4 \div 180) \times \sin(4)$
B. $(\pi \times 4 \div 180) \times \sin(\pi \times 4 \div 180)$
C. $4 \times \sin(\pi \times 4 \div 180)$
D. $(180 \times 4 \div \pi) \times \sin(4)$
E. $(180 \times 4 \div \pi) \times \sin(180 \times 4 \div \pi)$
F. $4 \times \sin(180 \times 4 \div \pi)$

Answer: F

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Question 10

1 mark

The real numbers

a

,

b

,

c

and

d

satisfy both

0 < a + b < c + d

and

0 < a + c < b + d

Which of the following inequalities must be true?
I

a < d

II

b < c

III

a + b + c + d > 0

A.none of them
B.I only
C.II only
D.III only
E.I and II only
F.I and III only
G.II and III only
H.I, II and III

Answer: F

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Question 11

1 mark

A spiral line is drawn as shown.
This spiral pattern continues indefinitely.
Which one of the following points is not on the spiral line?

A. $(99,100)$
B. $(99,-100)$
C. $(-99, 100)$
D. $(-99,-100)$
E. $(100,99)$
F. $(100, -99)$
G. $(-100,99)$
H. $(-100, -99)$

Answer: G

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Question 12

1 mark

Which one of A-F correctly completes the following statement?
Given that

a < b

, and

f(x) > 0

for all

x

with

a < x < b

, the trapezium rule produces an overestimate for

\int_a^b f(x)\,dx

...

A.... if $f'(x) > 0$ and $f''(x) < 0$ for all $x$ with $a < x < b$
B.... only if $f'(x) > 0$ and $f''(x) < 0$ for all $x$ with $a < x < b$
C.... if and only if $f'(x) > 0$ and $f''(x) < 0$ for all $x$ with $a < x < b$
D.... if $f'(x) < 0$ and $f''(x) > 0$ for all $x$ with $a < x < b$
E.... only if $f'(x) < 0$ and $f''(x) > 0$ for all $x$ with $a < x < b$
F.... if and only if $f'(x) < 0$ and $f''(x) > 0$ for all $x$ with $a < x < b$

Answer: D

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Question 13

1 mark

f(x)

is a function for which

\int_0^3 (f(x))^2\,dx + \int_0^3 f(x)\,dx = \int_0^1 f(x)\,dx

Which of the following claims about

f(x)

is/are necessarily true?
I

f(x) \le 0

for some

x

with

1\le x \le 3

II

\int_0^3 f(x)\,dx \le \int_0^1 f(x)\,dx

A.neither of them
B.I only
C.II only
D.I and II

Answer: D

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Question 14

1 mark

An arithmetic sequence

T

has first term

a

and common difference

d

, where

a

and

d

are non-zero integers.
Property P is:
For some positive integer

m

, the sum of the first

m

terms of the sequence is equal to the sum of the first

2m

terms of the sequence.
For example, when

a = 11

and

d = -2

, the sequence

T

has property P, because

11 + 9 + 7 + 5 = 11 + 9 + 7 + 5 + 3 + 1 + (-1) + (-3)

i.e. the sum of the first 4 terms equals the sum of the first 8 terms.
Which of the following statements is/are true?
I For

T

to have property P, it is sufficient that

ad < 0

.
II For

T

to have property P, it is necessary that

d

is even.

A.neither of them
B.I only
C.II only
D.I and II

Answer: A

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Question 15

1 mark

Which one of the following is a necessary and sufficient condition for

\sum_{k=1}^n \sin\left(\frac{k\pi}{3}\right) = \frac{\sqrt{3}}{2}

to be true?

A. $n=1$
B. $n$ is a multiple of 3
C. $n$ is a multiple of 6
D. $n$ is 1 more than a multiple of 3
E. $n$ is 1 more than a multiple of 6
F. $n$ is 1 more than a multiple of 6 or $n$ is 2 more than a multiple of 6

Answer: D

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Question 16

1 mark

The Fundamental Theorem of Calculus (FTC) tells us that for any polynomial

f

:

\frac{d}{dx}\left(\int_0^x f(t)\,dt\right) = f(x)

A student calculates

\frac{d}{dx}\left(\int_x^{2x} t^2\,dt\right)

as follows:
(I)

\int_x^{2x} t^2\,dt = \int_0^{2x} t^2\,dt - \int_0^x t^2\,dt

(II) By FTC,

\frac{d}{dx}\left(\int_0^x t^2\,dt\right) = x^2

(III) By FTC,

\frac{d}{dx}\left(\int_0^{2x} t^2\,dt\right) = (2x)^2 = 4x^2

(IV) So

\frac{d}{dx}\left(\int_x^{2x} t^2\,dt\right) = 4x^2 - x^2

(V) giving

\frac{d}{dx}\left(\int_x^{2x} t^2\,dt\right) = 3x^2

Which of the following best describes the student's calculation?

A.The calculation is completely correct.
B.The calculation is incorrect, and the first error occurs on line (I).
C.The calculation is incorrect, and the first error occurs on line (II).
D.The calculation is incorrect, and the first error occurs on line (III).
E.The calculation is incorrect, and the first error occurs on line (IV).
F.The calculation is incorrect, and the first error occurs on line (V).

Answer: D

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Question 17

1 mark

A set of six distinct integers is split into two sets of three.
The first set of three integers has a mean of 10 and a median of 8.
The second set of three integers has a mean of 12 and a median of 9.
What is the smallest possible range of the set of all six integers?

A.8
B.10
C.11
D.12
E.14
F.15

Answer: E

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Question 18

1 mark

In this question,

f(x) = ax^3 + bx^2 + cx + d

and

g(x) = px^3 + qx^2 + rx + s

are cubic polynomials.
If

f(x) - g(x) > 0

for every real

x

, which of the following is/are necessarily true?
I

a > p

II if

b = q

then

c = r

III

d > s

A.none of them
B.I only
C.II only
D.III only
E.I and II only
F.I and III only
G.II and III only
H.I, II and III

Answer: G

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Question 19

1 mark

Nine people are sitting in the squares of a 3 by 3 grid, one in each square, as shown.
Two people are called neighbours if they are sitting in squares that share a side.
(People in diagonally adjacent squares, which only have a point in common, are not called neighbours.)

Each of the nine people in the grid is either a truth-teller who always tells the truth, or a liar who always lies.
Every person in the grid says: 'My neighbours are all liars'.
Given only this information, what are the smallest number and the largest number of people who could be telling the truth?

A.smallest: 1, largest: 4
B.smallest: 2, largest: 4
C.smallest: 2, largest: 5
D.smallest: 3, largest: 4
E.smallest: 3, largest: 5
F.smallest: 4, largest: 4
G.smallest: 4, largest: 5
H.smallest: 5, largest: 5

Answer: E

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Question 20

1 mark

x

is a real number and

f

is a function.
Given that exactly one of the following statements is true, which one is it?

A. $x \ge 0$ only if $f(x) < 0$
B. $x < 0$ if $f(x) \ge 0$
C. $x \ge 0$ only if $f(x) \ge 0$
D. $f(x) < 0$ if $x < 0$
E. $f(x) \ge 0$ only if $x \ge 0$
F. $f(x) \ge 0$ if and only if $x < 0$

Answer: C

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